Characterizations and properties of graphs of Baire functions

Abstract

Abstract Let X be a paracompact topological space and Y be a Banach space. In this paper, we will characterize the Baire-1 functions f : X → Y by their graph: namely, we will show that f is a Baire-1 function if and only if its graph gr(f) is the intersection of a sequence ( G n ) n = 1 ∞ $\begin{array}{} \displaystyle (G_n)_{n=1}^{\infty} \end{array}$ of open sets in X × Y such that for all x ∈ X and n ∈ ℕ the vertical section of Gn is a convex set, whose diameter tends to 0 as n → ∞. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.